It is the presence of these harmonics that give each musical instrument
its peculiar sound. If we hear a trumpet, a clarinet and a violin
play the same note, we identify the source of the note by the blending
of the harmonics. The fundamental gives us the pitch, the harmonics
identify the source. The same is true of the various voices of the
pipe organ. The pipe builder causes certain harmonics to sound with
the fundamental by the shape of the pipe, the material from which it is
made and by using a particular scale or ratio (diameter to length).
Mutations and mixtures sound at the exact harmonics of the fundamental,
adding to the natural harmonics of the fundamental pitches.

These harmonics are what we listen for when we tune an instrument.

For example, if we wish to tune the 5th C - G, the second harmonic of C
and the first harmonic of G is G in the next higher octave. If our
interval is not tuned pure, you can hear a beat produced by the out-of-tune
harmonics. If it is pure, you will not hear a beat.

Let's look at how some of these intervals work together:

A pure 5th + a pure 4th = 1 octave.

If C' = 100, then 100 X 1 ½ = 150 (G') X 1 1/3 = 200 (C").

Trying to divide the octave in half or thirds doesn't work so well.

For example, 3 maj 3rds do not = an octave.

1.25 X 1.25 X 1.25 = 1.953125

To make division of the scale easier to understand, we use a logarithmic
system that divides the scale into 1200 cents per octave.

The elimination or accommodation of the comma is called tempering.
Tempering is an adjustment of the intervals between notes away from pure.
It is a compromise to make the intervals fit.

It is common to use the term 'tune' for both tuning and tempering.

'Tuning' is static and involves adjusting the pitch of one instrument to
another, as when a group of instrumentalists 'tune up'. It is also
the correct term to use when a string player 'tunes' the individual strings
of his instrument to each other.

When we 'tune', we adjust to unisons or pure intervals. These intervals
can be expressed as the ratio of integers.

'Tempering' is the correct term to use when we adjust pitches that are
not pure, but set to some other value. This would apply to fretted
string instruments and keyboard instruments.

In tempering, we distribute the comma over the 12 semitone intervals in
the octave.

These ntervals can only be expressed with irrational numbers.

The remainder of this discussion assumes we are dealing with a keyboard
instrument, specifically the organ.

The tempering of an organ is especially critical because:

Except for the reeds, the tone is quite pure - free of complex harmonics.

The volume of the tone does not decay, as on the piano.

The solution to the problem - tempering the scale

There are approximately 150 tempering schemes that have been advanced over
the centuries, with probably no more than 20 being practical for general
use. Of these, about 10 are practical for keyboard instruments, and
only one - equal temperament - practical for the modern piano.

Standard European 1/12 Diatonic Comma Equal Temperament

We will look at equal temperament first, even though it is considered a
modern temperament.

It is the most familiar.

It is the least difficult to understand because it divides into 12 equal
semitones.

It serves as a good model with which to compare the earlier temperaments.

There is evidence that equal temperament was known in China as early as
the 5th century, BC. It was introduced into western music early in
the 16th century with the invention of fretted string instruments.
It came into general use around 1854 in conjunction with the evolution
of the modern piano and the introduction of chromaticism and impressionism
in romantic music. Note that this date is 27 years after the death
of Beethoven!

In equal temperament, every 5th is narrow by 2 cents, (702 - 2 = 700),
and every 4th is wide by 2 cents, (498 + 2 - 500). This allows either
circle to close.

12 X 700 = 7 X 1200 = 8400.

12 X 500 = 5 X 1200 = 6000.

In the middle C octave, these 2-cent deviations cause the 4ths and 5ths
to beat at approximately 1 beat/sec.

In theory, all semitones are expressed as the 12th root of 2, which like
pi, is an irrational number. Several geometric and mathematical models
have been used to reach approximate values for each note. The ratio
of the semitone is slightly less than 18:17.

Enharmonic notes are the same pitch (C# and Db are the same note), and
3 equally tempered Maj 3rds DO equal an octave (3 X 400 = 1200).

Advantage of equal temperament:

You can play in any key. Every key is as in (or out) of tune as any
other.

Disadvantages of equal temperament:

There is no key color?

There is a great amount of tension as maj 3rds are very wide and min 3rds
are very narrow creating high beat rates (around 10/sec).

The high (and different) beat rates of the 3rds clash with the slow beat
rate of the 5th creating a very bad triad.

Our first written instructions for setting equal temperament come from
Giovanni Maria Lanfranco in 1533:

"The 5ths are tuned so flat that the ear is not well pleased with them;
and the 3rds are as sharp as can be endured."

Equal temperament has had its share of critics. Very few composers
or organists preferred equal temperament until the French Romantic school.

In 1879, William Pole wrote in his book The Philosophy of Music, "The modern
practice of tuning all organs to equal temperament has been a fearful detriment
to their quality of tone. Under the old tuning, an organ made harmonious
and attractive music. Now, the harsh 3rds give it a cacophonous and
repulsive effect."

In 1940, L. S. Lloyd wrote an article entitled The Myth of equal Temperament
in which he described the improbability of singers, or players of any instrument
with variable intonation of being able to sing or play in true equal temperament;
or, a keyboard instrument actually being tuned to theoretically correct
equal temperament.

Look at tuning scheme

Historical tuning systems

Just

Meantone

Well, or irregular

Just intonation

A family of temperaments based on pure octaves and pure 5ths - the first
2 harmonics

Used with pentatonic, diatonic and modal scales

Values for notes within the scale are mathematically derived.

First described by the Greek philosopher and mathematician, Pythagoras,
around 500 BC. Just intonation is often called Pythagorean tuning.

This tuning is not suitable for keyboard instruments. It is important
because most of the later systems are Pythagorean and it is the basis of
our diatonic (12-tone) scale.

In its basic form, it is a line of pure (or just) 5ths beginning with F:

F - C - G - D - A - E - B

Steps in the development of the Pythagorean scale, by 5ths and 4ths (inverted
5ths):

Create whole tone, Bb - C; and 2 semitones, A - Bb & Bb - B.
It was raised because it creates 2 semitones, rather than 1. It would
also have physically widened the octave which was mostly standardized by
the Middle Ages. Also the width of the whole tone, A - B would have
been half again as wide as the other whole tones.

Increased use of the chromatic scale, caused just intonation to become
impractical.

Meantone

Meantone tuning was developed during the Reformation.

Meantone is unique in that it is based on pure 3rds, rather than pure 5ths.

We earlier described the ditonic or Pythagorean comma, which was based
on a circle of 5ths and equals 24 cents. There is also a comma known
as the syntonic comma.

The interval by which 4 pure 5ths exceed 2 octaves plus a pure maj 3rd.
It is equal to 22 cents. This is the comma used in most meantone
temperaments.

Characteristics of meantone

D is exactly midway between C and E - at the mean - and is why it is called
meantone.

Ditonic semitones are larger than chromatic semitones (117 and 75 cents
in ¼ comma meantone), therefore enharmonic equivalents do not exist.
You must tune either a G# OR an Ab, for example, requiring re-tuning between
pieces.

White keys cannot be accidentals and double flats and double sharps do
not exist.

Because of the two previous items, it is classified as a restricted temperament.

The syntonic comma is distributed among the maj 3rds of the most remote
keys, making them quite unusable with one or more very poor intervals called
'wolves'. These wolves can be camouflaged with the use of trills
and other embellishments.

The best keys, C, F & G, also have pure, or nearly pure 4ths and 5ths
giving them a very still, peaceful quality.

The character of the sound becomes more lively as sharps or flats are added.

Earlier we discussed familiar measuring systems. There is one very
common measurement that we temper in an irregular manner - the calendar.
(discuss a 4-year period on the calendar - variable length of months and
the leap year)

Just intonation, meantone and equal temperament are classified as regular
systems because all, or all but one of the 5ths, are tempered equally.

Well temperament first appeared at about the same time as meantone, but
didn't come into general use until the time of Bach and Handel. Recent
scholarship has supported the proposition that this was the type of temperament
that Bach himself preferred, not equal temperament as advanced by 19th
century musicologists. His The Well Tempered Clavier was written
to demonstrate that music could be played in all keys.

Why well-tempering replaced meantone

Pre-Bach composers wrote to stay within the limits of meantone. They
also used its characteristics to their advantage by favoring harmonies
made up of 3rds and 6ths and using the poorer chords for contrast and affect.
They also wrote in modal keys to avoid the wolf intervals.

Composers could write chromatic music, but the performer had to tune the
raised keys to the correct intervals for the composition. This was
common in music written for the harpsichord, but music written for the
organ rarely exceeded 12 scale degrees - the raised keys always had the
same value.

One attempt at overcoming the limitations of meantone was to split the
raised keys with one part of the key playing an F# and the other part a
Gb, for example. It never gained acceptance because of increased
difficulty in construction, not to mention performance.

As Bach and others began writing music with more than 12 scale degrees,
meantone was no longer practical. Many of Bach's organ works contain
13 to 15 scale degrees and some go as high as 21. His entire keyboard
output uses 25 scale degrees, including a double Eb and a double C#.
Clearly, he had access to an instrument tuned to a newer temperament than
meantone.

Characteristics of well temperaments

All keys and chords are usable.

C major and A minor are normally the best keys; very still like meantone.
Movement increases as sharps or flats are added.

Most keys are better than equal temperament, with only the most remote
keys slightly worse.

Keys have characteristic colors.

C, F, and G are very peaceful and serene and are often used for pastorales
and other pieces of a quiet nature.

Keys with many sharps sound bright and cheerful and keys with many flats
sound somber and dark. The composers made good use of key colors.
Some temperaments have 3 or 4 groups of keys with similar colors, while
other temperaments gradually change as flats or sharps are added.

There are no wolf intervals.

The comma is distributed in an irregular manner, rather than more evenly
as in other systems.

Singers and instrumentalists have no problem adjusting to an irregular
temperament, as the actual pitches used are very close to those of equal
temperament.

The unevenness of the chromatic scale is not apparent.

Key modulation is available.

The raised keys are tempered to be enharmonic.

Irregular temperaments are modifications of one of the earlier tempering
systems. Most are Pythagorean in that they are based on the ditonic
comma and there are several pure 5ths.

All music sounds best when played in the temperament that the composer
was using at the time.

Tempering can be static, or change from moment to moment.

3 categories of instruments:

Those capable of playing any pitch and not confined to the 12 tone system
- tempering is continuous:

Violin family

Trombone

Human voice

Those whose construction provides mechanical or acoustic means for producing
the notes of the 12 tone system, but allow for minor changes in pitch by
the player. The instrument is tuned before performance and tempered
during performance:

Fretted string instruments

Valved brass instruments

Woodwinds

Those whose construction allows for pre-performance tuning/tempering, sometimes
by a technician, and no tempering during performance:

Most other instruments are just in nature and play flats slightly lower
than the enharmonic sharps.

The voice

Singers will easily adjust to the temperament of any instrument with which
they are singing.

An a cappella choir doesn't sing in any recognized temperament. Singers,
more so than string and woodwind players, constantly listen and make minute
adjustments in pitch to stay in tune with each other by singing all principal
intervals pure.

This constant tempering contributes to the tendency for unaccompanied singers
to go flat.

This is not so likely to occur if the music is written in a modal key,
or if the only chords are I, IV and V. It is very likely if there
is modulation and/or chromatic passages.

Meantone is best suited for the performance of early music on the harpsichord
or organ.

Equal temperament is the only practical temperament for the modern piano.

Large organs with romantic voicing and extensive unification should be
tuned to equal temperament.

Most other organs, especially those with baroque voicing, should be tuned
to a well temperament.